Discontinuous dynamics with grazing points

•The properties of discontinuous dynamics with grazing solutions are analyzed.•A variational system around a grazing solution is constructed.•The orbital stability of grazing cycles is examined analytically and numerically.•Small parameter method is extended for the analysis of grazing orbits.•The bifurcation of the grazing cycles is exemplified.

Discontinuous dynamical systems with grazing solutions are discussed. The group property, continuation of solutions, continuity and smoothness of solutions are thoroughly analyzed. A variational system around a grazing solution which depends on near solutions is constructed. Orbital stability of grazing cycles is examined by linearization. Small parameter method is extended for analysis of grazing orbits, and bifurcation of cycles is observed in an example. Linearization around an equilibrium grazing point is discussed. The results can be extended on functional differential equations, partial differential equations and others. Appropriate illustrations are depicted to support the theoretical results.